How Lagrangian states evolve into random waves

Ingremeau, Maxime; Rivera, Alejandro

Journal de l’École polytechnique - Mathématiques, Volume 9 (2022), pp. 177-212.

In this paper, we consider a compact connected manifold $(X,g)$ of negative curvature, and a family of semi-classical Lagrangian states $f_h\left(x\right)=a\left(x\right)e^{i\varphi \left(x\right)/h}$ on $X$. For a wide family of phases $\varphi$, we show that $f_h$, when evolved by the semi-classical Schrödinger equation during a long time, resembles a random Gaussian field. This can be seen as an analogue of Berry’s random waves conjecture for Lagrangian states.

Dans cet article, nous considérons une variété riemannienne connexe, compacte, de courbure sectionnelle négative, et une famille d’états lagrangiens semi-classiques $f_h\left(x\right)=a\left(x\right)e^{i\varphi \left(x\right)/h}$ sur $X$. Pour une grande famille de phases $\varphi$, nous montrons que $f_h$ que l’on fait évoluer par l’équation de Schrödinger pendant un temps long ressemble à un champ aléatoire gaussien. Ceci peut être vu comme un analogue de la conjecture des ondes aléatoires de Berry pour les états lagrangiens.

Classification: 35P99, 81B50, 81Q20
Keywords: Quantum chaos, semi-classical analysis, Berry’s conjecture, random waves
Mot clés : Chaos quantique, analyse semi-classique, conjecture de Berry, ondes aléatoires

[1] Abert, Miklos; Bergeron, Nicolas; Le Masson, Etienne Eigenfunctions and random waves in the Benjamini-Schramm limit, 2018 | arXiv

[2] Anantharaman, Nalini Entropy and the localization of eigenfunctions, Ann. of Math. (2), Volume 168 (2008) no. 2, pp. 435-475 | DOI | MR | Zbl

[3] Anantharaman, Nalini; Nonnenmacher, Stéphane Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold, Ann. Inst. Fourier (Grenoble), Volume 57 (2007) no. 7, pp. 2465-2523 http://aif.cedram.org/item?id=AIF_2007__57_7_2465_0 | DOI | Numdam | MR | Zbl

[4] Berry, M. V. Regular and irregular semiclassical wavefunctions, J. Phys. A, Volume 10 (1977) no. 12, pp. 2083-2091 http://stacks.iop.org/0305-4470/10/2083 | DOI | MR | Zbl

[5] Billingsley, Patrick Probability and measure, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2012

[6] Bourgain, Jean On toral eigenfunctions and the random wave model, Israel J. Math., Volume 201 (2014) no. 2, pp. 611-630 | DOI | MR | Zbl

[7] Buckley, Jeremiah; Wigman, Igor On the number of nodal domains of toral eigenfunctions, Ann. Inst. H. Poincaré Phys. Théor., Volume 17 (2016) no. 11, pp. 3027-3062 | DOI | MR | Zbl

[8] Cohen, Serge; Istas, Jacques Fractional fields and applications, Mathématiques & Applications, 73, Springer, Heidelberg, 2013 | DOI

[9] Dimassi, Mouez; Sjöstrand, Johannes Spectral asymptotics in the semi-classical limit, London Math. Soc. Lect. Note Series, 268, Cambridge University Press, Cambridge, 1999 | DOI

[10] Eberlein, Patrick Geodesic flows in manifolds of nonpositive curvature, Smooth ergodic theory and its applications (Seattle, WA, 1999) (Proc. Sympos. Pure Math.), Volume 69, American Mathematical Society, Providence, RI, 2001, pp. 525-571 | DOI | MR | Zbl

[11] Evans, Lawrence C. Partial differential equations, Graduate Studies in Math., 19, American Mathematical Society, Providence, RI, 2010 | DOI

[12] Golubitsky, M.; Guillemin, V. Stable mappings and their singularities, Graduate Texts in Math., 14, Springer-Verlag, New York-Heidelberg, 1973 | DOI

[13] Ingremeau, Maxime Local weak limits of Laplace eigenfunctions, Tunis. J. Math., Volume 3 (2021) no. 3, pp. 481-515 | arXiv | DOI

[14] Jost, Jürgen Riemannian geometry and geometric analysis, Universitext, Springer, Cham, 2017 | DOI

[15] Laptev, A.; Sigal, I. M. Global Fourier integral operators and semiclassical asymptotics, Rev. Math. Phys., Volume 12 (2000) no. 5, pp. 749-766 | DOI | MR | Zbl

[16] Nazarov, F.; Sodin, M. Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions, Zh. Mat. Fiz. Anal. Geom., Volume 12 (2016) no. 3, pp. 205-278 | DOI | MR | Zbl

[17] Nonnenmacher, Stéphane; Zworski, Maciej Quantum decay rates in chaotic scattering, Acta Math., Volume 203 (2009) no. 2, pp. 149-233 | DOI | MR | Zbl

[18] Romaniega, Á.; Sartori, Andrea Solutions to the Helmoltz equation satisfying the random wave model (work in progress)

[19] Sartori, Andrea Planck-scale number of nodal domains for toral eigenfunctions, J. Funct. Anal., Volume 279 (2020) no. 8, 108663, 21 pages | DOI | MR | Zbl

[20] Schubert, Roman Semiclassical behaviour of expectation values in time evolved Lagrangian states for large times, Comm. Math. Phys., Volume 256 (2005) no. 1, pp. 239-254 | DOI | MR | Zbl

[21] Zworski, Maciej Semiclassical analysis, Graduate Studies in Math., 138, American Mathematical Society, Providence, RI, 2012 | DOI